Calcmathlete

2 years ago

Let \(f\) and g be defined by\[f(x) = -x^2 + 8x - 4~~~~~1 ≤ x ≤ 6\]\[g(x) = 5 - x~~~~~0 ≤ x ≤ 7\]How would I go about with finding the domains of \(f(g(x))\) and \(g(f(x))\)? Also, how would I prove that \(f(g(3))\) is defined and \(g(f(3))\) is undefined? More than likely, it has to do with the domain, but I'm not fully sure with how to do it.

Well I now know how to find the domain of it if I wanted to, but this is a no calculator question, so it'd be hard to gauge the domain. I think I get it now because this is not true, it is undefined.
\[0 ≤ 11 ≤ 7\]